Worst case binary searchI'm not sure I understand, what is the data set? Is it all of [0,1]?if so it is infinite so we need to talk about that, binary search is defined for a finite collection. Also, it seems like you're thinking of searching in [0,1] with this method of eliminating halves of the interval each time, you can consider the binary representation of a number to estimate the number of steps needed for such a method

How to calculate $\log \log \log N$?Well, how do you calculate log(N)? What is the time complexity of that? Why do you think that calculating your expression by calculating each logarithm separately is slow?